Equivalence of Definitions of Transitive Closure (Relation Theory)/Union of Compositions is Smallest

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Theorem

Let $\RR$ be a relation on a set $S$.

Let:

$\RR^n := \begin{cases} \RR & : n = 0 \\ \RR^{n - 1} \circ \RR & : n > 0 \end{cases}$

where $\circ$ denotes composition of relations.



Finally, let:

$\displaystyle \RR^+ = \bigcup_{i \mathop \in \N} \RR^i$


Then $\RR^+$ is the smallest transitive relation on $S$ that contains $\RR$.


Proof

$\RR^+$ is Transitive

By Relation contains Composite with Self iff Transitive, we can prove that $\RR^+$ is transitive by proving the following:

$\RR^+ \circ \RR^+ \subseteq \RR^+$


Let $\tuple {a, c} \in \RR^+ \circ \RR^+$.

Then:

$\exists b \in S: \tuple {a, b} \in \RR^+, \tuple {b, c} \in \RR^+$

Thus:

$\exists n \in \N: \tuple {a, b} \in \RR^n$
$\exists m \in \N: \tuple {b, c} \in \RR^m$

From Composition of Relations is Associative:

$\RR^{n + m} = \RR^n \circ \RR^m$

so:

$\tuple {a, c} \in \RR^{n + m} \subseteq \RR^+$

Since this holds for all $\tuple {a, c} \in \RR^+ \circ \RR^+$:

$\RR^+ \circ \RR^+ \subseteq \RR^+$

Thus $\RR^+$ is transitive.

$\Box$


$\RR^+$ contains $\RR$

$\RR \subseteq \RR^+$ by Set is Subset of Union.


$\RR^+$ is Smallest

Let $\RR'$ be a transitive relation on $S$ such that $\RR \subseteq \RR'$.

We must show that $\RR^+ \subseteq \RR'$.

Let $\tuple {a, b} \in \RR^+$.

That is:

$a \mathrel \RR b$

Then:

$\exists n \in \N: \tuple {a, b} \in \RR^n$

Thus by the definition of composition of relations, there exists $x_{n-1} \in S$ such that:

$a \mathrel {\RR^{n - 1} } x_{n - 1} \land x_{n - 1} \mathrel \RR b$

Likewise there exists $x_{n-2} \in S$ such that:

$a \mathrel {\RR^{n - 2} } x_{n - 2} \land x_{n - 2} \mathrel \RR x_{n - 1}$

And so forth there exist elements $x_0, \dots, x_n \in S$ such that:

$x_0 = a$
$x_n = b$
$\forall k \in \N_n: x_k \mathrel \RR x_{k + 1}$

Since $\RR \subseteq \RR'$:

$\forall k \in \N_n: x_k \mathrel {\RR'} x_{k + 1}$

Since $\RR'$ is transitive:

$a \mathrel {\RR'} b$

That is:

$\tuple {a, b} \in \RR'$

Since this holds for all $\tuple {a, b} \in \RR^+$:

$\RR^+ \subseteq \RR'$

Since this holds for all transitive relations $\RR'$ that contain $\RR$:

$\RR^+$ is the smallest transitive relation containing $\RR$.

$\blacksquare$